L-functions and Multiplicative Number Theory

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The investigator and his colleagues study various problems in the analytic theory of L-functions and mean values of multiplicative functions. In particular the investigator aims for an improved understanding of the distribution of zeros of L-functions, their behaviour at special values, and the size of extreme values in the critical strip. The second main aim of this project is to describe the spectrum of possible mean-values of multiplicative functions whose values at primes are constrained to lie in special subsets of the unit disc. The investigator, in collaboration with Granville, intends to develop further the close connection between such mean values and solutions to a family of integral equations. This project concerns problems in the area of multiplicative number theory. A fundamental problem in this area is to understand the zeros of the Riemann zeta function and other L-functions. These functions encode much information about prime numbers, and other arithmetic objects. Besides being of intrinsic interest to mathematicians, ideas emerging from these problems have found use in computer science, cryptography and coding theory.